Math::Complex(3pm) Perl Programmers Reference Guide Math::Complex(3pm)
NAME
Math::Complex - complex numbers and associated mathematical functions
SYNOPSIS
use Math::Complex;
$z = Math::Complex->make(5, 6);
$t = 4 - 3*i + $z;
$j = cplxe(1, 2*pi/3);
DESCRIPTION
This package lets you create and manipulate complex numbers. By default, Perl limits itself to real
numbers, but an extra "use" statement brings full complex support, along with a full set of mathemat-ical mathematical
ical functions typically associated with and/or extended to complex numbers.
If you wonder what complex numbers are, they were invented to be able to solve the following equa-tion: equation:
tion:
x*x = -1
and by definition, the solution is noted i (engineers use j instead since i usually denotes an inten-sity, intensity,
sity, but the name does not matter). The number i is a pure imaginary number.
The arithmetics with pure imaginary numbers works just like you would expect it with real numbers...
you just have to remember that
i*i = -1
so you have:
5i + 7i = i * (5 + 7) = 12i
4i - 3i = i * (4 - 3) = i
4i * 2i = -8
6i / 2i = 3
1 / i = -i
Complex numbers are numbers that have both a real part and an imaginary part, and are usually noted:
a + bi
where "a" is the real part and "b" is the imaginary part. The arithmetic with complex numbers is
straightforward. You have to keep track of the real and the imaginary parts, but otherwise the rules
used for real numbers just apply:
(4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
(2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
A graphical representation of complex numbers is possible in a plane (also called the complex plane,
but it's really a 2D plane). The number
z = a + bi
is the point whose coordinates are (a, b). Actually, it would be the vector originating from (0, 0)
to (a, b). It follows that the addition of two complex numbers is a vectorial addition.
Since there is a bijection between a point in the 2D plane and a complex number (i.e. the mapping is
unique and reciprocal), a complex number can also be uniquely identified with polar coordinates:
[rho, theta]
where "rho" is the distance to the origin, and "theta" the angle between the vector and the x axis.
There is a notation for this using the exponential form, which is:
rho * exp(i * theta)
where i is the famous imaginary number introduced above. Conversion between this form and the carte-sian cartesian
sian form "a + bi" is immediate:
a = rho * cos(theta)
b = rho * sin(theta)
which is also expressed by this formula:
z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
In other words, it's the projection of the vector onto the x and y axes. Mathematicians call rho the
norm or modulus and theta the argument of the complex number. The norm of "z" will be noted abs(z).
The polar notation (also known as the trigonometric representation) is much more handy for performing
multiplications and divisions of complex numbers, whilst the cartesian notation is better suited for
additions and subtractions. Real numbers are on the x axis, and therefore theta is zero or pi.
All the common operations that can be performed on a real number have been defined to work on complex
numbers as well, and are merely extensions of the operations defined on real numbers. This means they
keep their natural meaning when there is no imaginary part, provided the number is within their defi-nition definition
nition set.
For instance, the "sqrt" routine which computes the square root of its argument is only defined for
non-negative real numbers and yields a non-negative real number (it is an application from R+ to R+).
If we allow it to return a complex number, then it can be extended to negative real numbers to become
an application from R to C (the set of complex numbers):
sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
It can also be extended to be an application from C to C, whilst its restriction to R behaves as
defined above by using the following definition:
sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
Indeed, a negative real number can be noted "[x,pi]" (the modulus x is always non-negative, so
"[x,pi]" is really "-x", a negative number) and the above definition states that
sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
which is exactly what we had defined for negative real numbers above. The "sqrt" returns only one of
the solutions: if you want the both, use the "root" function.
All the common mathematical functions defined on real numbers that are extended to complex numbers
share that same property of working as usual when the imaginary part is zero (otherwise, it would not
be called an extension, would it?).
A new operation possible on a complex number that is the identity for real numbers is called the con-jugate, conjugate,
jugate, and is noted with a horizontal bar above the number, or "~z" here.
z = a + bi
~z = a - bi
Simple... Now look:
z * ~z = (a + bi) * (a - bi) = a*a + b*b
We saw that the norm of "z" was noted abs(z) and was defined as the distance to the origin, also
known as:
rho = abs(z) = sqrt(a*a + b*b)
so
z * ~z = abs(z) ** 2
If z is a pure real number (i.e. "b == 0"), then the above yields:
a * a = abs(a) ** 2
which is true ("abs" has the regular meaning for real number, i.e. stands for the absolute value).
This example explains why the norm of "z" is noted abs(z): it extends the "abs" function to complex
numbers, yet is the regular "abs" we know when the complex number actually has no imaginary part...
This justifies a posteriori our use of the "abs" notation for the norm.
OPERATIONS
Given the following notations:
z1 = a + bi = r1 * exp(i * t1)
z2 = c + di = r2 * exp(i * t2)
z = <any complex or real number>
the following (overloaded) operations are supported on complex numbers:
z1 + z2 = (a + c) + i(b + d)
z1 - z2 = (a - c) + i(b - d)
z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
z1 ** z2 = exp(z2 * log z1)
~z = a - bi
abs(z) = r1 = sqrt(a*a + b*b)
sqrt(z) = sqrt(r1) * exp(i * t/2)
exp(z) = exp(a) * exp(i * b)
log(z) = log(r1) + i*t
sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.
The definition used for complex arguments of atan2() is
-i log((x + iy)/sqrt(x*x+y*y))
The following extra operations are supported on both real and complex numbers:
Re(z) = a
Im(z) = b
arg(z) = t
abs(z) = r
cbrt(z) = z ** (1/3)
log10(z) = log(z) / log(10)
logn(z, n) = log(z) / log(n)
tan(z) = sin(z) / cos(z)
csc(z) = 1 / sin(z)
sec(z) = 1 / cos(z)
cot(z) = 1 / tan(z)
asin(z) = -i * log(i*z + sqrt(1-z*z))
acos(z) = -i * log(z + i*sqrt(1-z*z))
atan(z) = i/2 * log((i+z) / (i-z))
acsc(z) = asin(1 / z)
asec(z) = acos(1 / z)
acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))
sinh(z) = 1/2 (exp(z) - exp(-z))
cosh(z) = 1/2 (exp(z) + exp(-z))
tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))
csch(z) = 1 / sinh(z)
sech(z) = 1 / cosh(z)
coth(z) = 1 / tanh(z)
asinh(z) = log(z + sqrt(z*z+1))
acosh(z) = log(z + sqrt(z*z-1))
atanh(z) = 1/2 * log((1+z) / (1-z))
acsch(z) = asinh(1 / z)
asech(z) = acosh(1 / z)
acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
arg, abs, log, csc, cot, acsc, acot, csch, coth, acosech, acotanh, have aliases rho, theta, ln,
cosec, cotan, acosec, acotan, cosech, cotanh, acosech, acotanh, respectively. "Re", "Im", "arg",
"abs", "rho", and "theta" can be used also as mutators. The "cbrt" returns only one of the solu-tions: solutions:
tions: if you want all three, use the "root" function.
The root function is available to compute all the n roots of some complex, where n is a strictly pos-itive positive
itive integer. There are exactly n such roots, returned as a list. Getting the number mathematicians
call "j" such that:
1 + j + j*j = 0;
is a simple matter of writing:
$j = ((root(1, 3))[1];
The kth root for "z = [r,t]" is given by:
(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
You can return the kth root directly by "root(z, n, k)", indexing starting from zero and ending at n
- 1.
The spaceship comparison operator, <=>, is also defined. In order to ensure its restriction to real
numbers is conform to what you would expect, the comparison is run on the real part of the complex
number first, and imaginary parts are compared only when the real parts match.
CREATION
To create a complex number, use either:
$z = Math::Complex->make(3, 4);
$z = cplx(3, 4);
if you know the cartesian form of the number, or
$z = 3 + 4*i;
if you like. To create a number using the polar form, use either:
$z = Math::Complex->emake(5, pi/3);
$x = cplxe(5, pi/3);
instead. The first argument is the modulus, the second is the angle (in radians, the full circle is
2*pi). (Mnemonic: "e" is used as a notation for complex numbers in the polar form).
It is possible to write:
$x = cplxe(-3, pi/4);
but that will be silently converted into "[3,-3pi/4]", since the modulus must be non-negative (it
represents the distance to the origin in the complex plane).
It is also possible to have a complex number as either argument of the "make", "emake", "cplx", and
"cplxe": the appropriate component of the argument will be used.
$z1 = cplx(-2, 1);
$z2 = cplx($z1, 4);
The "new", "make", "emake", "cplx", and "cplxe" will also understand a single (string) argument of
the forms
2-3i
-3i
[2,3]
[2,-3pi/4]
[2]
in which case the appropriate cartesian and exponential components will be parsed from the string and
used to create new complex numbers. The imaginary component and the theta, respectively, will
default to zero.
The "new", "make", "emake", "cplx", and "cplxe" will also understand the case of no arguments: this
means plain zero or (0, 0).
DISPLAYING
When printed, a complex number is usually shown under its cartesian style a+bi, but there are legiti-mate legitimate
mate cases where the polar style [r,t] is more appropriate. The process of converting the complex
number into a string that can be displayed is known as stringification.
By calling the class method "Math::Complex::display_format" and supplying either "polar" or "carte-sian" "cartesian"
sian" as an argument, you override the default display style, which is "cartesian". Not supplying any
argument returns the current settings.
This default can be overridden on a per-number basis by calling the "display_format" method instead.
As before, not supplying any argument returns the current display style for this number. Otherwise
whatever you specify will be the new display style for this particular number.
For instance:
use Math::Complex;
Math::Complex::display_format('polar');
$j = (root(1, 3))[1];
print "j = $j\n"; # Prints "j = [1,2pi/3]"
$j->display_format('cartesian');
print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
The polar style attempts to emphasize arguments like k*pi/n (where n is a positive integer and k an
integer within [-9, +9]), this is called polar pretty-printing.
For the reverse of stringifying, see the "make" and "emake".
CHANGED IN PERL 5.6
The "display_format" class method and the corresponding "display_format" object method can now be
called using a parameter hash instead of just a one parameter.
The old display format style, which can have values "cartesian" or "polar", can be changed using the
"style" parameter.
$j->display_format(style => "polar");
The one parameter calling convention also still works.
$j->display_format("polar");
There are two new display parameters.
The first one is "format", which is a sprintf()-style format string to be used for both numeric parts
of the complex number(s). The is somewhat system-dependent but most often it corresponds to "%.15g".
You can revert to the default by setting the "format" to "undef".
# the $j from the above example
$j->display_format('format' => '%.5f');
print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
$j->display_format('format' => undef);
print "j = $j\n"; # Prints "j = -0.5+0.86603i"
Notice that this affects also the return values of the "display_format" methods: in list context the
whole parameter hash will be returned, as opposed to only the style parameter value. This is a
potential incompatibility with earlier versions if you have been calling the "display_format" method
in list context.
The second new display parameter is "polar_pretty_print", which can be set to true or false, the
default being true. See the previous section for what this means.
USAGE
Thanks to overloading, the handling of arithmetics with complex numbers is simple and almost trans-parent. transparent.
parent.
Here are some examples:
use Math::Complex;
$j = cplxe(1, 2*pi/3); # $j ** 3 == 1
print "j = $j, j**3 = ", $j ** 3, "\n";
print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
$z = -16 + 0*i; # Force it to be a complex
print "sqrt($z) = ", sqrt($z), "\n";
$k = exp(i * 2*pi/3);
print "$j - $k = ", $j - $k, "\n";
$z->Re(3); # Re, Im, arg, abs,
$j->arg(2); # (the last two aka rho, theta)
# can be used also as mutators.
ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
The division (/) and the following functions
log ln log10 logn
tan sec csc cot
atan asec acsc acot
tanh sech csch coth
atanh asech acsch acoth
cannot be computed for all arguments because that would mean dividing by zero or taking logarithm of
zero. These situations cause fatal runtime errors looking like this
cot(0): Division by zero.
(Because in the definition of cot(0), the divisor sin(0) is 0)
Died at ...
or
atanh(-1): Logarithm of zero.
Died at...
For the "csc", "cot", "asec", "acsc", "acot", "csch", "coth", "asech", "acsch", the argument cannot
be 0 (zero). For the logarithmic functions and the "atanh", "acoth", the argument cannot be 1 (one).
For the "atanh", "acoth", the argument cannot be "-1" (minus one). For the "atan", "acot", the argu-ment argument
ment cannot be "i" (the imaginary unit). For the "atan", "acoth", the argument cannot be "-i" (the
negative imaginary unit). For the "tan", "sec", "tanh", the argument cannot be pi/2 + k * pi, where
k is any integer. atan2(0, 0) is undefined, and if the complex arguments are used for atan2(), a
division by zero will happen if z1**2+z2**2 == 0.
Note that because we are operating on approximations of real numbers, these errors can happen when
merely `too close' to the singularities listed above.
ERRORS DUE TO INDIGESTIBLE ARGUMENTS
The "make" and "emake" accept both real and complex arguments. When they cannot recognize the argu-ments arguments
ments they will die with error messages like the following
Math::Complex::make: Cannot take real part of ...
Math::Complex::make: Cannot take real part of ...
Math::Complex::emake: Cannot take rho of ...
Math::Complex::emake: Cannot take theta of ...
BUGS
Saying "use Math::Complex;" exports many mathematical routines in the caller environment and even
overrides some ("sqrt", "log", "atan2"). This is construed as a feature by the Authors, actually...
;-)
All routines expect to be given real or complex numbers. Don't attempt to use BigFloat, since Perl
has currently no rule to disambiguate a '+' operation (for instance) between two overloaded entities.
In Cray UNICOS there is some strange numerical instability that results in root(), cos(), sin(),
cosh(), sinh(), losing accuracy fast. Beware. The bug may be in UNICOS math libs, in UNICOS C com-piler, compiler,
piler, in Math::Complex. Whatever it is, it does not manifest itself anywhere else where Perl runs.
AUTHORS
Daniel S. Lewart <d-lewart@uiuc.edu>
Original authors Raphael Manfredi <Raphael_Manfredi@pobox.com> and Jarkko Hietaniemi <jhi@iki.fi>
perl v5.8.8 2001-09-21 Math::Complex(3pm)
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